On the boundedness of quaternionic numerical ranges with respect to nonstandard involutions

Let phi be a nonstandard involution on the set of all quaternion numbers, and alpha be a quaternion such that phi(alpha) = alpha. For any n x n quaternion matrix A, the notion of numerical range of A with respect to phi, W-phi((alpha) )(A), was introduced by L. Rodman in 2014, and for the case alpha = 0, he characterized all quaternion matrices A whose W-phi((0)) (A) is bounded. In this paper, for the case alpha not equal 0, those quaternion matrices A for which W-phi((alpha)) (A) is bounded, are characterized. Also, by introducing the notion of phi-class of a quaternion number, all quaternion matrices A whose W-phi((alpha)) (A) is a singleton set, are characterized, and for n >= 3, it is shown that W-phi((alpha)) (A) is bounded if and only if it is a singleton set. (C) 2020 Elsevier Inc. All rights reserved.